One-dimensional Chern-Simons theory and the $\hat{A}$ genus

TL;DR

This paper develops a one-dimensional Chern-Simons gauge theory for dg Lie and L-infinity algebras, quantizes it using BV formalism, and relates its partition function to the A-genus of a manifold, connecting topological quantum mechanics with derived geometry.

Contribution

It constructs a novel Chern-Simons theory for dg Lie and L-infinity algebras on 1-manifolds and links its partition function to the A-genus via derived geometric methods.

Findings

Partition function equals the A-genus of the manifold.

Quantization produces a volume form on the derived loop space.

Connects topological quantum mechanics with derived geometry.

Abstract

We construct a Chern-Simons gauge theory for dg Lie and L-infinity algebras on any one-dimensional manifold and quantize this theory using the Batalin-Vilkovisky formalism and Costello's renormalization techniques. Koszul duality and derived geometry allow us to encode topological quantum mechanics, a nonlinear sigma model of maps from a 1-manifold into a cotangent bundle T*X, as such a Chern-Simons theory. Our main result is that the partition function of this theory is naturally identified with the A-genus of X. From the perspective of derived geometry, our quantization construct a volume form on the derived loop space which can be identified with the A-class.